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eta-orthogonality and orthogonal complements in a Lorentz vector space

Definition

Let $(V, η)$ be a Lorentz vector space of signature $(-, +, +, +)$ (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space). Two vectors $u, v \in V$ are $η$ -orthogonal, written $u ⊥ v$ , when $η (u, v) = 0$ . For a subset $S \subseteq V$ the orthogonal complement of $S$ is

$S^{⊥} : = {v \in V : η (v, s) = 0 for every s \in S},$

and for a vector $u$ one writes $u^{⊥} : = {u}^{⊥}$ .

Elementary properties. For every subset $S \subseteq V$ :

$S^{⊥}$ is a linear subspace of $V$ : if $v, w \in S^{⊥}$ and $λ \in R$ , then for every $s \in S$ bilinearity gives $η (v + λ w, s) = η (v, s) + λ η (w, s) = 0$ , so $v + λ w \in S^{⊥}$ . In particular the zero vector lies in $S^{⊥}$ and $S^{⊥}$ is never empty.
$S \subseteq (S^{⊥})^{⊥}$ : if $s \in S$ , then $η (v, s) = 0$ for every $v \in S^{⊥}$ by the definition of $S^{⊥}$ , and symmetry of $η$ gives $η (s, v) = 0$ for every $v \in S^{⊥}$ , that is, $s \in (S^{⊥})^{⊥}$ .
$S^{⊥} = (span S)^{⊥}$ : the inclusion $S \subseteq span S$ gives $(span S)^{⊥} \subseteq S^{⊥}$ ; conversely, if $v \in S^{⊥}$ and $u = \sum_{i} λ_{i} s_{i}$ is a finite linear combination with $s_{i} \in S$ , then linearity in the second argument gives $η (v, u) = \sum_{i} λ_{i} η (v, s_{i}) = 0$ , so $v \in (span S)^{⊥}$ .

In particular, for every $u \in V$ one has $u \in u^{⊥}$ exactly when $η (u, u) = 0$ , that is, exactly when $u$ is null or $u = 0$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space).

A nonzero vector can be orthogonal to itself. Take a basis $e_{0}, e_{1}, e_{2}, e_{3}$ of $V$ in which the matrix of $η$ is diagonal with diagonal entries $- 1, 1, 1, 1$ ; such a basis exists by clause 2 of the definition of a Lorentz form. Then $v : = e_{0} + e_{1}$ is nonzero while $η (v, v) = η (e_{0}, e_{0}) + 2 η (e_{0}, e_{1}) + η (e_{1}, e_{1}) = - 1 + 0 + 1 = 0$ , so $v \in v^{⊥}$ with $v \neq 0$ . This is exactly what fails for a positive-definite inner product and is why orthogonal complements in a Lorentz vector space need not be positive definite.

eta-orthogonality and orthogonal complements in a Lorentz vector space

Definition

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